Predicting magic numbers of nuclei with semi-realistic nucleon-nucleon interactions

Magic numbers are predicted in wide range of the nuclear chart by the self-consistent mean-field calculations with the M3Y-P6 and P7 semi-realistic $NN$ interactions. The magic numbers are identified by vanishing pair correlations in the spherical Hartree-Fock-Bogolyubov regime. We also identify submagic numbers when energy gain due to the pairing is sufficiently small. It is found that the results with M3Y-P6 well correspond to the known data, apart from a few exceptions. For some of the magic or submagic numbers the prediction differs from that with the Gogny-D1S or D1M interaction. Roles of the tensor force and the spin-isospin channel originating from the one-pion exchange potential are investigated in the $Z$- or $N$-dependence of the shell gap.


Introduction
The shell structure is one of the fundamental concepts in nuclear physics, which is well described as if the constituent nucleons moved inside the nuclear mean field (MF) almost independently. A large energy gap between the single-particle (s.p.) orbits in the spherically symmetric MF produces a magic number, i.e. relative stability of nuclei having a specific proton (Z) or neutron number (N ), manifesting the shell structure. It has been disclosed by the experiments using the radioactive beams that the shell structure, and therefore the magic numbers, may depend on Z and N [1]. The magic numbers of nuclei are important in understanding the origin of matters, since elements were synthesized via nuclear reactions which should greatly be influenced by stability of relevant nuclei. For instance, several peaks in the abundance of elements correspond to the magic numbers of N . The magic numbers of nuclei could also be responsible for the existence limit of superheavy elements that have not yet been observed.
For the Z-and N -dependence of the shell structure, which is sometimes called shell evolution [2], several possible mechanisms have been pointed out. Since the height of the centrifugal barrier depends on the orbital angular momentum ℓ of the s.p. orbits, the s.p. energies of the loosely bound orbits may have significant ℓ-dependence, possibly influencing the shell evolution [3]. However, there has been no clear evidence that the ℓ-dependence of the s.p. energies gives rise to appearance or disappearance of magic numbers. On the other hand, roles of specific channels in the effective nucleonic interaction have been argued. It has been clarified that the tensor channels are important in the shell evolution [4,5,6]. The central spin-isospin channel of the nucleon-nucleon (N N ) interaction has also been considered [7]. In addition to the N N interaction, roles of the three-nucleon (N N N ) interaction have been investigated [8].
While significance of the tensor force in the shell evolution has been clarified qualitatively in Ref. [4], effects of specific channels on magic numbers should be assessed carefully, because in practice magic numbers emerge via interplay of various channels. One of the authors (H.N.) has developed the M3Ytype semi-realistic N N interactions in a series of articles [9,10,11,12]. By applying the numerical methods of Refs. [13,14,15], the self-consistent Hartree-Fock (HF) and Hartree-Fock-Bogolyubov (HFB) calculations have been implemented with the semi-realistic interactions. Since the interactions explicitly include the tensor force with realistic origin and have reasonable nature on the central spin-isospin channel, the MF approaches employing the semi-realistic interactions provide us with a suitable framework for investigating the magic numbers in wide range of the nuclear chart. A good example is the s.p. level inversion observed from 40 Ca to 48 Ca, which is reproduced remarkably well by the M3Y-type interactions including the realistic tensor force [6]. Note also that the numerical methods of Refs. [13,14,15] have ability to handle loosely bound orbitals. The MF calculations with the semi-realistic interactions have been implemented to investigate the shell structure in neutron-rich Ca and Ni nuclei [5]. In this paper we extensively apply the spherical MF approaches with the semi-realistic interactions, particularly the parameter-sets M3Y-P6 and P7 [12], and predict magic numbers in wide range of the nuclear chart, from relatively light to heavy nuclei including nuclei far off the β stability. As discarded in the conventional MF approaches, tensor-force effects are one of the current hot topics in nuclear structure physics. We investigate effects of the tensor force and those of the spin-isospin channel by comparing the results to those with the Gogny-D1S [16] and D1M [17] interactions.

Effective Hamiltonian
Throughout this paper the Hamiltonian is taken to be H = H N + V C − H c.m. , consisting of the nuclear Hamiltonian H N , the Coulomb interaction V C and the center-of-mass (c.m.) Hamiltonian H c.m. . The following non-relativistic form is assumed for H N , and where i and j are the indices of individual nucleons, M = (M p + M n )/2 with M p (M n ) representing the mass of a proton (a neutron) [18], s i is the spin operator of the i-th nucleon, r ij = r i − r j , r ij = |r ij |, p ij = (p i − p j )/2, L ij = r ij × p ij , and ρ(r) denotes the nucleon density. The tensor operator is defined by The P SE , P TE , P SO and P TO operators indicate the projection on the singlet-even (SE), triplet-even (TE), singlet-odd (SO) and triplet-odd (TO) two-particle states. The c.m. Hamiltonian is H c.m. = P 2 /2AM , where P = i p i and A = Z + N . In the M3Y-type interactions f (X) n is taken to be the Yukawa function for all of X = C, LS, TN. In the Gogny-D1S and D1M interactions we have f (C) n (r) = e −(µnr) 2 , f (LS) (r) = ∇ 2 δ(r) and v (TN) = 0. The density-dependent contact term v (DD) carries certain effects of the N N N interaction and the nuclear medium (as introduced via the G-matrix), which are significant to reproduce the saturation properties. It is emphasized that v (TN) is quite realistic in M3Y-P6 and P7, since it is kept unchanged from the M3Y-Paris interaction [19] which has the G-matrix origin. The longest-range part in v (C) is also realistic, maintained to be the corresponding term of the one-pion exchange potential (OPEP) and denoted by v OPEP , the M3Y-type interactions have spin-isospin properties consistent with the experimental information [12]. Adjusted to the microscopic results of the pure neutron matter [20,21], M3Y-P6 and P7 give reasonable symmetry energy up to its density dependence [12].

Identification of magic and submagic numbers
Definition of magic numbers is not necessarily clear. Experimentally, they have been identified by relative stability; e.g. mass irregularity, kink in the separation energies and high excitation energies. From theoretical viewpoints, the magic nature, which is called "magicity", is linked to quenching of the manybody correlations. Typically, the spherical HF solution is expected to give a good approximation for the doubly magic nuclei. The nuclei having either magic Z or N are usually spherical. For spherical nuclei, the pairing among like nucleons provides dominant correlation beyond the HF. We shall therefore identify magic numbers (and submagic numbers) by comparing the spherical HF and HFB results for even-even nuclei.
The quadrupole deformation can be another source that breaks magicity. However, both the pair excitation and the quadrupole deformation are driven by quenching of the shell gap in most cases, although strength of the residual interaction associated with the individual correlation is relevant as well. Thus the sizable pair correlation within the spherical HFB could also be a measure of the quadrupole deformation. Conversely, when we view breaking of magicity via the pairing in this study, it does not necessarily indicate that the breakdown takes place due to the pair correlation in actual. The present work would give informative overview of distribution of magic numbers over the nuclear chart. Although more precise investigation by taking into account the deformation degrees of freedom (d.o.f) will be desirable, we leave it as a future study which needs intensive calculations.
The magic numbers Z = 2, 8, 20, 28, 50, 82 and N = 2, 8, 20, 28, 50, 82, 126 have been established around the β stability line. These numbers are kept to be magic in certain region, whereas it has been clarified by experiments that the N = 8, 20, 28 magicity is eroded far off the β stability. As will be shown in Sec. 4, N = 184 can also be a good magic number, insensitive to Z. We first implement the HF and HFB calculations for even-even nuclei along these numbers. If we find a good candidate for another magic number, we additionally carry out calculations around it. Since validity of the MF approaches could be questioned in very light nuclei and it is difficult to collect experimental data for extremely heavy nuclei, we restrict ourselves to the nuclei having 8 ≤ Z ≤ 126 and N ≤ 200.
From the spherical HFB results, we identify Z (N ) to be magic when the proton (neutron) pair correlation vanishes. We recognize the vanishing pair correlation through the proton (neutron) pair energy E pair p (E pair n ), i.e. the energy contributed by the proton (neutron) pairing tensor in the HFB state. There are certain cases in which the pair correlation survives, but the HF and the HFB energies, which are denoted by E HF and E HFB , are very close. In these nuclei correlation effects are suppressed, resulting in e.g. relatively high excitation energy. Such suppression takes place particularly when either of Z or N is a good magic number, as has been pointed out for 68 Ni [5] and 146 Gd [22]. We therefore identify Z (N ) to be submagic if E HF − E HFB is smaller than a certain value λ sub for N = magic (Z = magic) nuclei. We adopt λ sub = 0.5 MeV and 0.8 MeV in the calculations below, independent of A. Comparison of the results using different λ sub values will exhibit how they are sensitive (or insensitive) to λ sub .
There could be disputes on the above criterion for the magic and submagic numbers. It is true that the criterion based solely on the spherical MF calculations is not complete. We may find influence of quadrupole deformation if comparing the current D1S results to those of the comprehensive deformed HFB calculations [23]. However, previous studies have suggested [5] that the semi-realistic interactions often give simple picture for appearance and disappearance of the magicity, in good connection to the shell structure under the spherical symmetry. Indeed, prediction of magicity with the semi-realistic interactions is in good harmony with the known data, particularly for relatively light-mass region. Moreover, since quadrupole deformation may quench magicity but cannot enhance it, the prediction within the spherical MF regime is useful in selecting candidates and in overviewing how the magic numbers can distribute over the nuclear chart.
There were several works in which the magicity was studied from kinks in the two-proton (S 2p ) or two-neutron (S 2n ) separation energies [24]. Whereas S 2p and S 2n are calculable within the spherical HFB calculations, it is not straightforward to draw a quantitative measure for magicity from S 2p and S 2n that is applicable to wide range of the nuclear chart. We here comment that the magic and submagic numbers shown below, which are identified from the criterion given above, are compatible with the kinks in the 3 calculated S 2p or S 2n .

Predicting magic and submagic numbers
We now show prediction on magic and submagic numbers, which are identified by the criterion given in Sec. 3. For the HF and HFB calculations, we have used the methods developed in Ref. [14] with the basis functions of Ref. [15]. The s.p. bases up to ℓ = ℓ max + 2, where ℓ max is the highest ℓ of the occupied level in the HF configuration, should be included for each nucleus to handle the pair correlation appropriately [14]. We use the ℓ ≤ 7 bases for the N < 82 nuclei, the ℓ ≤ 8 bases for the 82 ≤ N < 126 nuclei and the ℓ ≤ 9 bases for the N ≥ 126 nuclei in the HFB. No additional approximations are imposed, by explicitly treating the exchange and the pairing terms of V C [25] as well as the 2-body term of H c.m. .

Overview of magic and submagic numbers
To illustrate how we identify magic and submagic numbers, we show E pair    in E HF − E HFB , and we consider that N = 16 is submagic at 24 O in the M3Y-P6 result. Analogously, N = 14 is submagic both with D1M and M3Y-P6.
In Fig. 2, E pair n as well as E pair p are presented for the N = 28 isotones. We remark that non-vanishing E pair n is obtained in the Z ≤ 14 region with the M3Y-P6 interaction. The same holds with M3Y-P7, though not shown. This well corresponds to the observed breakdown of the N = 28 magicity in this region. At 42 Si E HF − E HFB is less than 0.8 MeV. However, deformation may be driven because neither Z nor N is magic. For this reason we do not regard N = 28 to be submagic at 42 Si, although deformed MF calculations are required for full justification. In contrast, when we apply D1M, Z = 14 is magic and N = 28 remains submagic at 42 Si, although the N = 28 magicity is lost at 40 Mg.
The N = 28 magicity holds in Z ≥ 16. We do not find quenching of the proton pairing in this region, except at the normal magic numbers Z = 20 and 28.
We search magic and submagic numbers in this manner in the wide range of the nuclear chart, applying the spherical HF and HFB calculations. The prediction with the M3Y-P6 and P7 semi-realistic interactions is summarized in Figs. 3 and 4. For comparison, the prediction with D1M is displayed in

N = 6, 14 and 16
The N = 16 magicity at 24 O observed in the experiments [3] is more or less reproduced by any of the four interactions; magic in the Gogny interactions (D1S and D1M) while submagic in the M3Y-P6 and P7 interactions. The N = 16 magicity is extended to Z = 10 with D1M and even to Z = 14 with D1S. We find that N = 14 as well as N = 6 behave magic or submagic in the O nuclei, as seen in Fig. 1.

N = 20 and 28
It is remarked that the N = 20 magicity is lost at Z = 10 with M3Y-P6 and at Z = 10, 12 with M3Y-P7, though not with D1M. This reminds us of the experimentally established 'island of inversion'. Usual interpretation of the island of inversion has been the deformation around N = 20 for the Z 12 nuclei [26], whereas possibility of the large fluctuation of the pairing has also been argued [27]. Although the present calculation does not tell us its mechanism, it is interesting to view that the loss of the magicity is well described with M3Y-P7, and partially with M3Y-P6.
Another notable point is that N = 20 becomes submagic at 40 Ca, rather than magic, with M3Y-P6 and M3Y-P7. This point should further be investigated in the near future.
As already argued in relation to

N = 32, 34 and 40
Although N = 32 and 34 are magic for the Si and the S isotopes with D1M (so is N = 34 with D1S), this prediction is not supported with the semi-realistic M3Y-P6 and P7 interactions. Irrespective of the interactions, N = 32 is submagic at 52 Ca owing to the closure of the n1p 3/2 orbit, as is consistent with the experimental data [28]. However, the N = 32 magicity is not found at 60 Ni with M3Y-P6 and P7, though kept with D1S and D1M. Origin of the Z-and the interaction-dependence of the N = 32 magicity will be discussed in Sec. 5. The recently suggested magicity of N = 34 at 54 Ca [29] is not seen in Figs. 3 -5. The submagic nature of N = 40 at 68 Ni, which is again compatible with the data [30], is reproduced by all the interactions. On the other hand, the N = 40 magicity is lost at 60 Ca in the M3Y-P6 and P7 results, while it is preserved in the D1S and D1M results. The N = 40 magicity depends on the interactions also at 80 Zr, to which we shall return in Sec. 4.10.

N = 50, 56 and 58
We do not find loss of the N = 50 magicity except at 70 Ca. It is noted that N = 50 is indicated to be submagic at 70 Ca, rather than magic, with the M3Y-P6 and P7 interactions. The 70 Ca nucleus is not bound within the spherical HFB with D1S and D1M.
We find that N = 56 is submagic at 96 Zr except with D1M because of the n1d 5/2 closure, well corresponding to the high first excitation energy in the measurements [31]. In the D1S and D1M results N = 56 and 58 are submagic in the Ni isotopes. Although the submagic nature of N = 58 has been argued using M3Y-P5 in Ref. [5], the magicity is not apparent in the M3Y-P6 and P7 results.

Z = 14 and 16
Having Z = 14 and N = 20, 34 Si is predicted to be doubly magic with any of the four interactions, as has been argued in connection to the proton bubble structure [6]. Z = 14 stays as a magic number in the results with the D1S and D1M interactions. In contrast, it is not a magic number in N ≥ 26 with M3Y-P6 and P7, nor at N = 14 with M3Y-P6. The stiffness of the Z = 14 core seems relevant to where the N = 28 magicity is broken, which has been argued in Sec. 4.3.
All the interactions indicate that Z = 16 is weakly submagic at 36 S. With the D1M interaction Z = 16 becomes magic at N = 12 and 14 (also at N = 16 with D1S), while it is not with the M3Y-P6 and P7 interactions.

Z = 20 and 28
In the present study, Z = 20 and 28 remain to be magic numbers in the whole region of N and irrespective of the effective interactions. As pointed out in Ref. [5], the persistence of the Z = 28 magicity around 78 Ni is contrasted to the argument in Ref. [4], although the realistic tensor force is included in M3Y-P6 and P7. This difference happens because magic numbers are a result of interplay of various interaction channels, even though the tensor force plays a significant role.

Z = 34, 38 and 40
The Z = 38 and 40 magicity has been known to be enhanced along the N = 50 isotones. This nature is well described, with both Z's staying submagic.
We find that Z = 34 is submagic with M3Y-P6 and P7 at N = 82 while 116 Se is unbound with D1S and D1M. Z = 38 is predicted to be submagic with all of the four interactions at 120 Sr.
The Zr isotopes have been known to exhibit remarkable N -dependence in their structure. While 90 Zr is close to doubly magic, the Zr nuclei are deformed in 60 ≤ N 70 as well as in N ≈ 40. We find that the current results for the Zr isotopes significantly depend on the interactions. Although Z = 40 is magic in 38 ≤ N ≤ 46 with D1M (in 38 ≤ N ≤ 48 with D1S), this magicity is broken with M3Y-P6 and P7, which is consistent with the experimental data. As a typical example, 80 Zr is doubly magic with the Gogny interactions, while both Z and N lose magicity with the M3Y-P6 and P7 interactions. The observed energy levels show that 80 Zr is deformed and never doubly magic. The deformed HFB does not solve this problem of the D1S interaction [23].
In the present work using the spherical MF calculations, Z = 40 is indicated to be magic in 60 N 70, with any of the four interactions. This is contradictory to the recent experiments [32]. It is desired to implement deformed MF calculations, particularly those using the semi-realistic interactions, to check whether and how the experimental data is accounted for.
The 122 Zr nucleus is doubly magic in the M3Y-P6 and P7 results, and is close to doubly magic with submagic nature of Z = 40 in the Gogny results.

Z = 50, 58 and 64
We predict no breakdown of the Z = 50 magicity in the present calculations. However, deformation has been suggested for neutron-rich Sn nuclei in the previous calculations; e.g. for 98 ≤ N ≤ 110 in the calculations with D1S [23] and similarly in the relativistic MF calculations [33]. Further study is desirable by applying the M3Y-P6 and P7 interactions to the deformed MF calculations.
All the current interactions reproduce the submagic nature of Z = 64 at 146 Gd, which has long been investigated (e.g. Ref. [22]). We also find that Z = 58 becomes submagic at 140 Ce with M3Y-P6 and P7, though not with D1S and D1M. The measured first excitation energy is slightly higher in 140 Ce [34]

Z = 82 and 92
Within the spherical MF calculations, the Pb nuclei are bound up to N = 184 with all the interactions, and the last bound nucleus 266 Pb is predicted to be a good doubly magic nucleus. Since quadrupole deformation has been predicted to take place in 144 ≤ N ≤ 166 by the deformed HFB calculations with D1S [23], stability against deformation should further be investigated in future studies.
The Z = 92 number gains certain magicity, behaves as submagic at 218 U. Whereas Z = 92 is not fully closed in the neutron-deficient region, it is predicted to be magic in N 150 in the present work. This magicity is stronger in the D1S, D1M and M3Y-P7 results, even holding at 238 U which is a well-known deformed nucleus [35], in contrast to the M3Y-P6 result in which Z = 92 is not magic up to N = 150. It should be mentioned that deformed ground states have been predicted by the deformed HFB calculations with D1S in N 170 [23].

Z = 120, 124 and 126
The proton magic numbers beyond Z = 100 have attracted interest, in connection to the superheavy nuclei in the so-called 'island of stability'. Although Z = 114 has been considered a candidate of a magic number, the present work does not support it irrespectively of the interactions. On the other hand, Z = 120 may behave as a magic number. We view that Z = 120 is magic in N ≤ 178 with any of the four interactions. In the M3Y-P7 result the Z = 120 magicity extends to N = 200, and in the M3Y-P6 result it disappears at N = 180 but revives at N = 196, apart from its the submagic nature at N = 184. The Z = 120 magicity is also predicted in N ≤ 182 and N ≥ 194 with D1S. In the D1M result the Z = 120 magicity is lost in N ≥ 180, despite its submagic nature at N = 184. Z = 124 is predicted to be submagic at N = 184 with D1S, M3Y-P6 and P7, while unbound within the spherical HFB with D1M. Z = 126 is a good magic number with D1S, M3Y-P6 and M3Y-P7, but not with D1M. It is commented that the fission d.o.f. may enter in the Z 120 nuclei [23]. OPEP in some detail, by analyzing the spherical HF results. Since these channels are explicitly contained in M3Y-P6 and P7 while not in D1S and D1M, comparison among these results will be useful in assessing effects of v (TN) and v (C) OPEP on the magicity. We here comment on the difference in the magicity between M3Y-P6 and P7, which is found mainly in heavier mass region in Figs. 3 and 4. As the v (TN) and v (C) OPEP channels are identical between them, it is not obvious what is the main source of the difference between M3Y-P6 and P7. Although these two parameter-sets yield different neutron-matter energies at high densities, it is not likely that this significantly influences the magicity of nuclei. The strength of v (LS) , which is slightly stronger in M3Y-P7 than in M3Y-P6, does not account for all the visible difference between Figs. 3 and 4.
The s.p. energy of the orbit j, ε τz (j) (τ z = p, n), is defined by the derivative of the total energy with respect to the occupation probability. We extract contribution of v (TN) to ε τz (j) from the full HF result by where n j denotes the occupation probability on j. Likewise for ε OPEP . For the shell gap ∆ε τz (j 2 -j 1 ) = ε τz (j 2 )−ε τz (j 1 ), corresponding quantities ∆ε can be considered. Since the s.p. energies are more or less fitted to the data in each effective interaction, the absolute values of ∆ε is important, which may give rise to the Z-or N -dependence of the shell gap. We shall argue several typical cases in which v (TN) and/or v OPEP play a significant role in Z-or N -dependence of the magicity.

N = 16, 32 and 40
While N = 16 behaves as magic or submagic at 24 O irrespectively of the effective interactions, its magicity depends on the interactions for larger Z, i.e. near the β stability. The shell gap ∆ε n (0d 3/2 − 1s 1/2 ) is relevant to the N = 16 magicity. In Ref. [10] we have shown that v (TN) and v (C) OPEP give the Z-dependence by using the older parameter-set M3Y-P5. Analogous results are obtained with the present parameters M3Y-P6 and P7, which we do not repeat here.

N = 56
From Figs. 3 -5 we have found that the submagic nature of N = 56 at 96 Zr is well accounted for with M3Y-P6 and P7, but not with D1S and D1M. We present the relevant s.p. energy difference ∆ε n (0g 7/2 -1d 5/2 ) in Fig. 6, for D1M and M3Y-P6. It is found that ∆ε n strongly depends on Z in the M3Y-P6 result, as p0g 9/2 is occupied in 40 ≤ Z ≤ 50. This Z-dependence produces the relatively large gap at 96 Zr. The M3Y-P6 result also suggests that n1d 5/2 and n0g 7/2 are nearly degenerate around 106 Sn, compatible with the observed levels in 105,107 Sn [36].
In order to clarify effects of v

N = 164
As shown in Figs. 3 -5 N = 164 becomes a submagic with M3Y-P6 and a magic number with M3Y-P7 at 256 U, although it is not with the D1S and D1M interactions. The N = 164 shell gap is primarily determined by ∆ε n (1g 7/2 -0j 15/2 ), and is relevant also to ∆ε n (2d 5/2 -0j 15/2 ). These s.p. energy differences are shown for D1M and M3Y-P6 in Fig. 7. We find enhancement of the gap at Z = 92 in the M3Y-P6 case, which is regarded as origin of the magicity. We do not have similar Z-dependence in the D1M result.
We find opposite trends between the D1M and the M3Y-P6 results on ∆ε p (0g 9/2 -1p 1/2 ) in 40 ≤ N ≤ 50 as n0g 9/2 is occupied, and in 70 ≤ N ≤ 82 as n0h 11/2 is occupied. In addition, the rising tendency of ∆ε p (0g 9/2 -1p 1/2 ) at N ≈ 60 in the M3Y-P6 result is not conspicuous with D1M. It is notable that, after contributions of v (C) OPEP and v (TN) are subtracted, the M3Y-P6 result becomes almost parallel to that of D1M. On the contrary, once v (TN) is set in, the s.p. energy difference has quite similar Ndependence to the full result. This clarifies significance of v (TN) in the N -dependence of the shell gap at Z = 40. Relatively small ∆ε p (0g 9/2 -1p 1/2 ) at N = 40 contributes to the loss of the Z = 40 magicity around 80 Zr viewed in Fig. 3, and relatively large ∆ε p (0g 9/2 -1p 1/2 ) at N = 82 to the persistence of the magicity around 122 Zr. The large ∆ε p (0g 9/2 -1p 1/2 ) at N ≈ 60 prevents the Z = 40 magicity from being broken within the spherical HFB. It will be an interesting future subject whether and how the observed deformation at N 60 is accounted for, under the sizable shell gap brought by v (TN) .

Z = 58
The Z = 58 magicity in the neutron-rich region of N 110 takes place because of the p0g 7/2 occupation, to which the energy difference ∆ε p (1d 5/2 -0g 7/2 ) is relevant. However, when comparing ∆ε p (1d 5/2 -0g 7/2 )  OPEP as in Fig. 6. The region where Z = 40 is magic in Fig. 5 (Fig. 3) is shown by the blue (red) arrows, and submagic by the skyblue (orange) circles at the top part of the figure.
between D1M and M3Y-P6, it should be noted that n0i 13/2 lies lower than n2p 3/2 in the spherical HF calculation with M3Y-P6 in 100 ≤ N ≤ 120, while these two orbits are inverted with D1M except at 172 Ce. We therefore show ∆ε p (1d 5/2 -0g 7/2 ) with D1M in which the s.p. levels were filled in the same ordering as in the M3Y-P6 case, by the blue dashed line in Fig. 10. If the neutron occupation is taken to be similar, ∆ε p (1d 5/2 -0g 7/2 ) with M3Y-P6 after removing the v (C) OPEP and v (TN) contributions is almost parallel to ∆ε p (1d 5/2 -0g 7/2 ) with D1M. As seen in Fig. 10, v (TN) gives rise to large ∆ε p (1d 5/2 -0g 7/2 ) at N ≈ 114, with a cooperative effect of v (C) OPEP . Although its degree depends on other channels of the interactions as recognized by comparing Figs. 3 and 4, it is expected that the Z = 58 magicity is enhanced in N 110 because of v (TN) and v apparently, except at 32 Mg and the Zr isotopes with 60 ≤ N 70. Although calculations including the deformation degrees of freedom are needed for complete understanding of the magic numbers, this work will be useful in selecting candidates of the magic numbers and in overviewing how the magic numbers can distribute over the nuclear chart.
By analyzing the shell gaps, roles of the tensor force and of the central spin-isospin channel from the OPEP are investigated. It is confirmed that the tensor force often plays a significant role in the Zor N -dependence of the shall gap, accounting for appearance and disappearance of magicity, and the central spin-isospin channel tends to enhance the tensor-force effect for emergence of the magicity. The present results are qualitatively consistent with Refs. [4,7], although quantitative aspects should not be underestimated because they make difference in certain cases.
It will be interesting to study effects of deformation on the magicity with the semi-realistic interactions, and to investigate whether and how the discrepancy between the current results and the data in several nuclei could be resolved.